Bunuel wrote:

SOLUTION

If 4<(7-x)/3, which of the following must be true?

I. 5<x

II. |x+3|>2

III. -(x+5) is positive

(A) II only

(B) III only

(C) I and II only

(D) II and III only

(E) I, II and III

Note that we are asked to determine which MUST be true, not could be true.

\(4<\frac{7-x}{3}\) --> \(12<7-x\) --> \(x<-5\). So we know that \(x<-5\), it's given as a fact. Now, taking this info we should find out which of the following inequalities will be true OR which of the following inequalities will be true for the range \(x<-5\).

Basically the question asks: if \(x<-5\) which of the following is true?

I. \(5<x\) --> not true as \(x<-5\).

II. \(|x+3|>2\), this inequality holds true for 2 cases, (for 2 ranges): 1. when \(x+3>2\), so when \(x>-1\) or 2. when \(-x-3>2\), so when \(x<-5\). We are given that second range is true (\(x<-5\)), so this inequality holds true.

Or another way: ANY \(x\) from the range \(x<-5\) (-5.1, -6, -7, ...) will make \(|x+3|>2\) true, so as \(x<-5\), then \(|x+3|>2\) is always true.

III. \(-(x+5)>0\) --> \(x<-5\) --> true.

Answer: D.

Bunuel if the question asks: if \(x<-5\) which of the following is true?

how can \(|x+3|>2\), this inequality hold true for BOTH cases ?

you write "when \(x+3>2\), so when \(x>-1\) "

so if \(x>-1\) how can it hold true when \(x<-5\) So if \(x < -5\) then \(x\) can be -6, -7 -8 , -9 etc all negative numbers starting from -6 , whereas \(x>-1\) means thar x can be 0, 1, 3, 4 etc all positive numbers

can you explain this part, please